Citing "personal and political" reasons, Varadkar, 45, announced his decision at a press conference in Dublin on Wednesday, saying in an at-times emotional speech that he no longer felt he was the "best person" to lead Ireland.
Earlier this month his government suffered damaging defeats in two referendums on references to family and women in the constitution.
Varadkar, who said he was resigning as party leader with immediate effect, is expected to be replaced as taoiseach as soon as his successor as party leader is able to take office.
While his departure will not automatically trigger a snap election, it comes only 10 weeks before European parliamentary and local elections and less than a year before Ireland's next general election.
Varadkar said: "One part of leadership is knowing when the time has come to pass on the baton, and then having the courage to do it. That time is now."
In a statement read on the steps of government buildings in the Irish capital, he said: "I believe this government can be re-elected ... I believe a new taoiseach will be better placed than me to achieve that - to renew and strengthen the top team, to refocus our message and policies and to drive implementation."
He added that he had asked for a new leader of the party to be chosen on 6 April, allowing a new prime minister and cabinet to be elected after parliament's Easter break.
While he was "deeply grateful" for his time in office and "would wholeheartedly recommend a career in politics", Varadkar said he had reached the end of the road as taoiseach. "Politicians are human beings and we have our limitations," he said. "We give it everything until we can't anymore. And then we have to move on."
Contenders to succeed him as Fine Gael leader and new prime minister include the higher education minister, Simon Harris, who is the bookmakers' clear favourite; the enterprise minister and former deputy PM, Simon Coveney; the public expenditure minister, Paschal Donohoe, and the justice minister, Helen McEntee.
Varadkar said his reasons for stepping down were "mainly political" but did not elaborate on what they were. Earlier this month, he was widely blamed for a crushing twin defeat at the ballot box, including the biggest ever referendum loss by an Irish government.
The ruling coalition had proposed rewording the 1937 constitution to change outdated references to family and women. Critics said Varadkar had rushed the debate in a "gimmicky" effort to hold double referendums on International Women's Day, and accused him of presiding over "incoherent messaging".
Voters rejected the family referendum, with 67% voting no, and buried the other proposal, which related to women's care-giving role, in an even bigger landslide of 74%. Varadkar later accepted some responsibility, saying: "There are a lot of people who got this wrong and I am certainly one of them."
Until Wednesday's resignation, however, the political fallout from the debacle had widely been expected to be limited. Varadkar has also faced increasing discontent within Fine Gael, with 10 of its members of the Dáil Éireann - almost a third of the party's total - announcing they will not stand again at the next election, which must be called by early 2025.
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“I need to stop fantasizing about running away to some other life and start figuring out the one I have.”
Holly Black, The Darkest Part of the Forest